Gravitational lensing strong weak and micro

2002 discussquasar lensing by galaxies and provide an intuitive geometrical optics ap-proach to lensing, Fort and Mellier 1994 describe the giant luminous arcsand arclets in clusters of

P Schneider C Kochanek J Wambsganss

Gravitational Lensing: Strong, Weak and Micro Saas-Fee Advanced Course 33

Swiss Society for Astrophysics and Astronomy

Edited by G Meylan, P Jetzer and P North

With 196 Illustrations, 36 in Color

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who, with his two colleagues Bob Carswell and Ray Weymann, discovered in 1979 the first extragalactic gravitational lens, the

quasar QSO 0957+0561

The observation, in 1919 by A.S Eddington and collaborators, of the tational deflection of light by the Sun proved one of the many predictions ofEinstein’s Theory of General Relativity: The Sun was the first example of agravitational lens.

gravi-In 1936, Albert Einstein published an article in which he suggested ing stars as gravitational lenses A year later, Fritz Zwicky pointed out thatgalaxies would act as lenses much more likely than stars, and also gave a list

us-of possible applications, as a means to determine the dark matter content us-ofgalaxies and clusters of galaxies

It was only in 1979 that the first example of an extragalactic gravitationallens was provided by the observation of the distant quasar QSO 0957+0561,

by D Walsh, R.F Carswell, and R.J Weymann A few years later, the firstlens showing images in the form of arcs was detected

The theory, observations, and applications of gravitational lensing tute one of the most rapidly growing branches of astrophysics The gravita-tional deflection of light generated by mass concentrations along a light pathproduces magnification, multiplicity, and distortion of images, and delays pho-ton propagation from one line of sight relative to another The huge amount

consti-of scientific work produced over the last decade on gravitational lensing hasclearly revealed its already substantial and wide impact, and its potential forfuture astrophysical applications

The 33rd Saas-Fee Advanced Courses of the Swiss Society for Astronomy

and Astrophysics, entitled Gravitational Lensing: Strong, Weak, and Micro,

took place from 8–12 April, 2003, in Les Diablerets, a pleasant mountain resort

of the Swiss Alps The three lecturers were Peter Schneider, Christopher S.Kochanek, and Joachim Wambsganss

These proceedings are provided in four complementary parts of a book on

gravitational lensing P Schneider wrote Part 1, Introduction to Gravitational

Lensing and Cosmology, the first draft of which was made available to all

registered participants a week before the course C.S Kochanek wrote Part 2

about Strong Gravitational Lensing, while P Schneider in Part 3 dealt with

Weak Gravitational Lensing, and J Wambsganss in Part 4 about Gravitational Microlensing.

We are thankful to Nicole Tharin, the secretary of the Laboratoired’Astrophysique de l’Ecole Polytechnique F´ed´erale de Lausanne (EPFL), forher continuous presence and efficient help, and to Yves Debernardi for hisefficient logistic support during the course We are equally thankful to Fr´ed´ericCourbin, Dominique Sluse, Christel Vuissoz, and Alexander Eigenbrod for help

in the editorial process of this book

The meeting was also sponsored by the Universit´e de Lausanne, the EcolePolytechnique F´ed´erale de Lausanne (EPFL), the Swiss Society for Astron-omy and Astrophysics, the Acad´emie Suisse des Sciences Naturelles, the FondsNational Suisse de la Recherche Scientifique, the Space Telescope ScienceInstitute, the Universit¨at Z¨urich, and the Observatoire de Gen`eve

Pierre North

Part 1: Introduction to Gravitational Lensing and Cosmology

P Schneider 1

1 Introduction 1

1.1 History of Gravitational Light Deflection 2

1.2 Discoveries 5

1.3 What is Lensing Good for? 14

2 Gravitational Lens Theory 18

2.1 The Deflection Angle 18

2.2 The Lens Equation 20

2.3 Magnification and Distortion 23

2.4 Critical Curves and Caustics, and General Properties of Lenses 25

2.5 The Mass-Sheet Degeneracy 29

3 Simple Lens Models 31

3.1 Axially Symmetric Lenses 31

3.2 The Point-Mass Lens 34

3.3 The Singular Isothermal Sphere 36

3.4 Non-Symmetric Lenses 38

4 The Cosmological Standard Model I: The Homogeneous Universe 44

4.1 The Cosmic Expansion 44

4.2 Distances and Volumes 49

4.3 Gravitational Lensing in Cosmology 52

5 Basics of Lensing Statistics 54

5.1 Cross-Sections 55

5.2 Lensing Probabilities; Optical Depth 57

5.3 Magnification Bias 58

6 The Cosmological Standard Model II: The Inhomogeneous Universe 61

6.1 Structure Formation 61

6.2 Halo Abundance and Profile 71

6.3 The Concordance Model 77

6.4 Challenges 81

7 Final Remarks 83

References 84

Part 2: Strong Gravitational Lensing C S Kochanek 91

1 Introduction 91

2 An Introduction to the Data 92

3 Basic Principles 97

3.1 Some Nomenclature 98

3.2 Circular Lenses 101

3.3 Non-Circular Lenses 112

4 The Mass Distributions of Galaxies 121

4.1 Common Models for the Monopole 125

4.2 The Effective Single Screen Lens 129

4.3 Constraining the Monopole 130

4.4 The Angular Structure of Lenses 136

4.5 Constraining Angular Structure 140

4.6 Model Fitting and the Mass Distribution of Lenses 143

4.7 Non-Parametric Models 150

4.8 Statistical Constraints on Mass Distributions 152

4.9 Stellar Dynamics and Lensing 158

5 Time Delays 163

5.1 A General Theory of Time Delays 165

5.2 Time Delay Lenses in Groups or Clusters 169

5.3 Observing Time Delays and Time Delay Lenses 170

5.4 Results: The Hubble Constant and Dark Matter 174

5.5 The Future of Time Delay Measurements 181

6 Gravitational Lens Statistics 182

6.1 The Mechanics of Surveys 182

6.2 The Lens Population 185

6.3 Cross Sections 192

6.4 Optical Depth 193

6.5 Spiral Galaxy Lenses 196

6.6 Magnification Bias 197

6.7 Cosmology With Lens Statistics 205

6.8 The Current State 206

7 What Happened to the Cluster Lenses? 210

7.1 The Effects of Halo Structure and the Power Spectrum 216

7.2 Binary Quasars 218

8 The Role of Substructure 221

8.1 Low Mass Dark Halos 230

9 The Optical Properties of Lens Galaxies 232

9.1 The Interstellar Medium of Lens Galaxies 238

10 Extended Sources and Quasar Host Galaxies 243

10.1 An Analytic Model for Einstein Rings 243

10.2 Numerical Models of Extended Lensed Sources 248

10.3 Lensed Quasar Host Galaxies 251

11 Does Strong Lensing Have a Future? 255

References 256

Part 3: Weak Gravitational Lensing P Schneider 269

1 Introduction 269

2 The Principles of Weak Gravitational Lensing 272

2.1 Distortion of Faint Galaxy Images 272

2.2 Measurements of Shapes and Shear 274

2.3 Tangential and Cross Component of Shear 277

2.4 Magnification Effects 280

3 Observational Issues and Challenges 281

3.1 Strategy 282

3.2 Data Reduction: Individual Frames 284

3.3 Data Reduction: Coaddition 288

3.4 Image Analysis 292

3.5 Shape Measurements 295

4 Clusters of Galaxies: Introduction, and Strong Lensing 298

4.1 Introduction 298

4.2 General Properties of Clusters 299

4.3 The Mass of Galaxy Clusters 301

4.4 Luminous Arcs and Multiple Images 304

4.5 Results from Strong Lensing in Clusters 309

5 Mass Reconstructions from Weak Lensing 315

5.1 The Kaiser–Squires Inversion 316

5.2 Improvements and Generalizations 317

5.3 Inverse Methods 324

5.4 Parameterized Mass Models 327

5.5 Problems of Weak Lensing Cluster Mass Reconstruction and Mass Determination 330

5.6 Results 333

5.7 Aperture Mass and Other Aperture Measures 343

5.8 Mass Detection of Clusters 346

6 Cosmic Shear – Lensing by the LSS 355

6.1 Light Propagation in an Inhomogeneous Universe 356

6.2 Cosmic Shear: The Principle 358

6.3 Second-Order Cosmic Shear Measures 360

6.4 Cosmic Shear and Cosmology 366

6.5 E-Modes, B-Modes 371

6.6 Predictions; Ray-Tracing Simulations 377

7 Large-Scale Structure Lensing: Results 382

7.1 Early Detections of Cosmic Shear 383

7.2 Integrity of the Results 384

7.3 Recent Cosmic Shear Surveys 386

7.4 Detection of B-Modes 392

7.5 Cosmological Constraints 394

7.6 3-D Lensing 397

7.7 Discussion 400

8 The Mass of, and Associated with Galaxies 404

8.1 Introduction 404

8.2 Galaxy–Galaxy Lensing 405

8.3 Galaxy Biasing: Shear Method 416

8.4 Galaxy Biasing: Magnification Method 427

9 Additional Issues in Cosmic Shear 430

9.1 Higher-Order Statistics 430

9.2 Influence of LSS Lensing on Lensing by Clusters and Galaxies 436

10 Concluding Remarks 439

References 442

Part 4: Gravitational Microlensing J Wambsganss 453

1 Lensing of Single Stars by Single Stars 454

1.1 Brief History 454

1.2 Theoretical Background 454

1.3 How Good is the Point Lens – Point Source Approximation? 458

1.4 Statistical Ensembles 460

2 Binary Lenses 461

2.1 Theory and Basics of Binary Lensing 462

2.2 First Microlensing Lightcurve of a Binary Lens: OGLE-7 466

2.3 Binary Lens MACHO 1998-SMC-1 467

2.4 Binary Lens MACHO 1999-BLG-047 471

2.5 Binary Lens EROS BLG-2000-005 472

3 Microlensing and Dark Matter: Ideas, Surveys and Results 475

3.1 Why We Need Dark Matter: Flat Rotation Curves (1970s) 475

3.2 How to Search for Compact Dark Matter (as of 1986) 477

3.3 Just Do It: MACHO, EROS, OGLE et al (as of 1989) 477

3.4 “Pixel”-Lensing: Advantage Andromeda! 478

3.5 Current Interpretation of Microlensing Surveys with Respect to Halo Dark Matter (as of 2004) 479

3.6 Microlensing toward the Galactic Bulge 484

4 Microlensing Surveys in Search of Extrasolar Planets 486

4.1 How Does the Microlensing Search for Extrasolar Planet Work? The Method 486

4.2 Why Search for Extrasolar Planets with Microlensing? –

Advantages and Disadvantages 488

4.3 Who is Searching? The Teams: OGLE, MOA, PLANET, MicroFUN 492

4.4 What is the Status of Microlensing Planet Searches so far? The Results 493

4.5 When will Planets be Detected with Microlensing? The Prospects 496

4.6 Note Added in April 2004 (About One Year after the 33rd Saas Fee Advanced Course) 497

4.7 Summary 497

5 Higher Order Effects in Microlensing: 499

6 Astrometric Microlensing 516

7 Quasar Microlensing 520

7.1 Microlensing Mass, Length and Time Scales 521

7.2 Early and Recent Theoretical Work on Quasar Microlensing 524

7.3 Observational Evidence for Quasar Microlensing 526

7.4 Quasar Microlensing: Now and Forever? 534

References 536

Index 541

!! 2004 The Sun, Solar Analogs and the Climate

J.D Haigh, M Lockwood, M.S Giampapa

!! 2003 GravitationalLensing: Strong,Weak and Micro

P Schneider, C Kochanek, J Wambsganss

!! 2002 The Cold Universe

A.W Blain, F Combes, B.T Draine

!! 2001 Extrasolar Planets

T Guillot, P Cassen, A Quirrenbach

!! 2000 High-Energy Spectroscopic Astrophysics

S.M Kahn, P von Ballmoos, R.A Sunyaev

!! 1999 Physics of Star Formation in Galaxies

F Palla, H Zinnecker

!! 1998 Star Clusters

B.W Carney, W.E Harris

!! 1997 Computational Methods for Astrophysical Fluid Flow

R.J LeVeque, D Mihalas, E.A Dorfi, E M¨ uller

!! 1996 Galaxies Interactions and Induced Star Formation

R.C Kennicutt, F Schweizer, J.E Barnes

!! 1995 Stellar Remnants

S.D Kawaler, I Novikov, G Srinivasan

* 1994 Plasma Astrophysics

J.G Kirk, D.B Melrose, E.R Priest

* 1993 The Deep Universe

A.R Sandage, R.G Kron, M.S Longair

* 1992 Interacting Binaries

S.N Shore, M Livio, E.J.P van den Heuvel

* 1991 The Galactic Interstellar Medium

W.B Burton, B.G Elmegreen, R Genzel

* 1990 Active Galactic Nuclei

R Blandford, H Netzer, L Woltjer

! 1989 The Milky Way as a Galaxy

G Gilmore, I King, P van der Kruit

! 1988 Radiation in Moving Gaseous Media

H Frisch, R.P Kudritzki, H.W Yorke

! 1987 Large Scale Structures in the Universe

A.C Fabian, M Geller, A Szalay

! 1986 Nucleosynthesis and Chemical Evolution

J Audouze, C Chiosi, S.E Woosley

R.S Booth, J.W Brault, A Labeyrie

! 1984 Planets, Their Origin, Interior and Atmosphere

D Gautier, W.B Hubbard, H Reeves

! 1983 Astrophysical Processes in Upper Main Sequence Stars

A.N Cox, S Vauclair, J.P Zahn

* 1982 Morphology and Dynamics of Galaxies

J Binney, J Kormendy, S.D.M White

! 1981 Activity and Outer Atmospheres of the Sun and Stars

F Praderie, D.S Spicer, G.L Withbroe

* 1980 Star Formation

J Appenzeller, J Lequeux, J Silk

* 1979 Extragalactic High Energy Physics

F Pacini, C Ryter, P.A Strittmatter

* 1978 Observational Cosmology

J.E Gunn, M.S Longair, M.J Rees

* 1977 Advanced Stages in Stellar Evolution

I Iben Jr., A Renzini, D.N Schramm

* 1976 Galaxies

K Freeman, R.C Larson, B Tinsley

* 1975 Atomic and Molecular Processes in Astrophysics

A Dalgarno, F Masnou-Seeuws, R.V.P McWhirter

* 1974 Magnetohydrodynamics

L Mestel, N.O Weiss

* 1973 Dynamical Structure and Evolution of Stellar Systems

G Contopoulos, M H´ enon, D Lynden-Bell

* 1972 Interstellar Matter

N.C Wickramasinghe, F.D Kahn, P.G Metzger

* 1971 Theory of the Stellar Atmospheres

D Mihalas, B Pagel, P Souffrin

Lensing and Cosmology

P Schneider

1 Introduction

Light rays are deflected when they propagate through a gravitational field.Long suspected before General Relativity – the theory which we believe pro-vides the correct description of gravity – it was only after Einstein’s finalformulation of this theory that the effect was described quantitatively Therich phenomena which are caused by this gravitational light deflection has led

to the development of the rather recent active research field of gravitationallensing, and the fact that the 2003 Saas-Fee course is entirely devoted to thissubject is just but one of the indications of the prominence this topic hasachieved In fact, the activities in this area have become quite diverse andare reflected by the three main lectures of this course The phenomena oflight propagation in strong gravitational fields, as it occurs near the surface

of neutron stars or black holes, are usually not incorporated into tional lensing – although the physics is the same, these strong-field effectsrequire a rather different mathematical description than the weak deflectionphenomena

gravita-In this introductory first part (PART 1) we shall provide an outline ofthe basics of gravitational lensing, covering aspects that are at the base of

it and which will be used extensively in the three main lectures We start inSect 1.1 with a brief historical account; the study of the influence of a gravita-tional field on the propagation of light started long before the proper theory ofgravity – Einstein’s General Relativity – was formulated Illustrations of themost common phenomena of gravitational lensing will be given next, before

we will introduce in Sect 2 the basic equations of gravitational lensing theory

A few simple lens models will be considered in Sect 3, in particular the mass lens and the singular isothermal sphere model Since the sources anddeflectors in gravitational lensing are often located at distances comparable

point-to the radius of the observable Universe, the large-scale geometry of time needs to be accounted for Thus, in Sect 4 we give a brief introduction tothe standard model of cosmology We then proceed in Sect 5 with some basicSchneider P (2006), Introduction to gravitational lensing and cosmology In: Meylan G, Jetzer Ph and North P (eds) Gravitational lensing: Strong, weak, and micro Saas-Fee Adv Courses vol 33, pp 1–89

space-DOI 10.1007/3-540-30309-X 1  Springer-Verlag Berlin Heidelberg 2006c

considerations about lensing statistics, i.e., the question of how probable it isthat observations of a source at large distance are significantly affected by alensing effect, and conclude with a description of the large-scale matter distri-bution in the Universe The material covered in this introductory part will beused extensively in the later parts of this book; those will be abbreviated as SL(Strong Lensing, Kochanek, 2005, Part 2 of this book), WL (Weak Lensing,Schneider, 2005, Part 3 of this book), and ML (MicroLensing, Wambsganss,

2005 Part 4 of this book)

Gravitational lensing as a whole, and several particular aspects of it, hasbeen reviewed previously Two extensive monographs (Schneider et al 1992,hereafter SEF; Petters, Levine and Wambsganss 2001, hereafter PLW) de-scribe lensing in great detail, in particular providing a derivation of the gravi-tational lensing equations from General Relativity (see also Seitz et al 1994).Blandford and Narayan (1992) review the cosmological applications of gravi-tational lensing, Refsdall and Surdej (1994) and Courbin et al (2002) discussquasar lensing by galaxies and provide an intuitive geometrical optics ap-proach to lensing, Fort and Mellier (1994) describe the giant luminous arcsand arclets in clusters of galaxies, Paczy´nski (1996) reviews the effects of grav-itational microlensing in the local group, the review by Narayan and Bartel-mann (1999) provides a concise account of gravitational lensing theory andobservations, and Mellier (1999), Bartelmann and Schneider (2001), Wittman(2002) and van Waerbeke and Mellier (2003) review the relatively young field

of weak gravitational lensing

1.1 History of Gravitational Light Deflection

We start with a (very) brief account on the history of gravitational lensing;the reader is referred to SEF and PLW for a more detailed presentation

The Early Years, Before General Relativity

The Newtonian theory of gravitation predicts that the gravitational force F on

a particle of mass m is proportional to m, so that the gravitational acceleration

a = F/m is independent of m Therefore, the trajectory of a test particle in

a gravitational field is independent of its mass but depends, for a given initialposition and direction, only on the velocity of the test particle About 200years ago, several physicists and astronomers speculated that, if light could

be treated like a particle, light rays may be influenced in a gravitational field

as well John Mitchell in 1784, in a letter to Henry Cavendish, and laterJohann von Soldner in 1804, mentioned the possibility that light propagating

in the field of a spherical mass M (like a star) would be deflected by an angle

ˆ

αN = 2GM/(c2ξ), where G and c are Newton constant of gravity and the

velocity of light, respectively, and ξ is the impact parameter of the incoming

light ray At roughly the same time, Pierre-Simon Laplace in 1795 noted “thatthe gravitational force of a heavenly body could be so large, that light could

not flow out of it” (Laplace 1975), i.e., that the escape velocity ve=

2GM/R from the surface of a spherical mass M of radius R becomes the velocity of light, which happens if R = Rs≡ 2GM/c2, nowadays called the Schwarzschild

radius of a mass M

Gravitational Light Deflection in GR

All these results were derived under the assumption that light somehow can

be considered like a massive test particle; this was of course well before theconcept of photons was introduced Only after the formulation of GeneralRelativity by Albert Einstein in 1915 could the behavior of light in a gravita-tional field be studied on a firm physical ground Before the final formulation

of GR, Einstein published a paper in 1911 where he recalculated the results

of Mitchell and Soldner (of whose work he was unaware) for the deflectionangle Only after the completion of GR did it become clear that the ‘New-tonian’ value of the deflection angle was too small by a factor of 2 In thegeneral theory of relativity, the deflection is

in 1919, with a sufficient accuracy to distinguish between the ‘Newtonian’and the GR value, provided a tremendous success for Einstein’s new theory

of gravity

Soon thereafter, Lodge (1919) used the term ‘lens’ in the context of itational light deflection, but noted that ‘it has no focal length’ Chwolson(1924) considered a source perfectly coaligned with a foreground mass, con-cluding that the source should be imaged as a ring around the lens – in fact,only fairly recently did it become known that Einstein made some unpublishednotes on this effect in 1912 (Renn et al 1997) – hence, calling them ‘Einsteinrings’ is indeed appropriate If the alignment is not perfect, two images of thebackground source would be visible, one on either side of the foreground star.Einstein, in 1936, after being approached by the Czech engineer Rudi Mandl,wrote a paper where he considered this lensing effect by a star, including boththe image positions, their separation, and their magnifications He concludedthat the angular separation between the two images would be far too small(of order milli-arcseconds) to be resolvable, so that “there is no great chance

grav-of observing this phenomenon” (Einstein 1936)

Zwicky’s Visions

This pessimistic view was not shared by Fritz Zwicky , who in 1937 publishedtwo truly visionary papers Instead of looking at lensing by stars in our Galaxy,

he considered “extragalactic nebulae” (nowadays called galaxies) as lenses –with his mass estimates of these nebulae, he estimated typical image separa-tion of a background source to be of order 10– about one order of magnitude

too high – and such pairs of images can be separated with telescopes

Observ-ing such an effect, he noted, would furnish an additional test of GR, allowone to see galaxies at larger distances (due to the magnification effect), andwould determine the masses of these nebulae acting as lenses (Zwicky 1937a)

He then went on to estimate the probability that a distant source would belensed to produce multiple images, concluded that about 1 out of 400 distantsources should be affected by lensing (this is about the fractional area covered

by the bright parts of nebulae on photographic plates), and hence predictedthat “the probability that nebulae which act as gravitational lenses will befound becomes practically a certainty” (Zwicky 1937b) As we shall see in duecourse, basically all of Zwicky’s predictions became true.1

The Revival of Lensing

Until the beginning of the 1960’s the subject rested, but in 1963/4, threeauthors independently reopened the field: Klimov (1963), Liebes (1964) andRefsdal (1964a,b) Klimov considered lensing of galaxies by galaxies, whereasLiebes and Refsdal mainly studied lensing by point-mass lenses Their pa-pers have been milestones in lensing research; for example, Liebes consideredthe possibility that stars in the Milky Way can act as lenses for stars inM31 – we shall see in ML, this is a truly modern idea Refsdal calculatedthe difference of the light travel times between the two images of a source –since light propagates along different paths from the source to the observer,there will in general be a time delay which can be observed provided thesource is variable, such like a supernova Refsdal pointed out that the timedelay depends on the mass of the lens and the distances to the lens andthe source, and concluded that, if the image separation and the time delaycould be measured, the lens mass and the Hubble constant could be deter-mined We shall see in SL (Part 2) how these predictions have been realized inthe meantime

In 1963, the first quasars were detected: luminous, compact (‘quasi-stellar’)and very distant sources – hence, a source population had been discoveredwhich lies behind Zwicky’s nebulae, and finding lens systems amongst themshould be a certainty Nevertheless, it took another 15 years until the firstlens system was observed and identified as such

1.2 Discoveries

First Detections of Multiple Imaging (1979)

In their program to optically identify radio sources, Walsh et al in 1979 covered a pair of quasars separated by about 6 arcseconds, having identical

dis-colors, redshifts (zs= 1.41) and spectra (see Walsh 1989 for the history of this

discovery) The year 1979 also marked two important technical developments

in astronomy: the first CCD detectors replaced photographic plates, thus viding much higher sensitivity, dynamic range and linearity, and the very largearray (VLA), a radio interferometer providing radio images of subarcsecondimage quality, went into operation With the VLA it was soon demonstratedthat both quasar images are compact radio sources, with similar radio spec-tra Soon thereafter, a galaxy situated between the two quasar images wasdetected (Stockton 1980; Young et al 1980) The galaxy has a redshift of

pro-zd = 0.36 and it is the brightest galaxy in a small cluster We now know

that the cluster contributes its share to the large image separation in this tem Furthermore, the first very long baseline interferometry (VLBI) data ofthis system, known as QSO 0957+561, showed that both components have acore-jet structure with the symmetry expected for lensed images of a commonsource (see Fig 1) The great similarities of the two optical spectra (Fig 2) isanother proof of the lensing nature of this system

sys-One year later, the so-called triple quasar PG 1115+080 was discovered(Weymann et al 1980) It apparently consisted of three images, one of whichwas much brighter than the other two (see Fig 3) Soon thereafter it was shownthat the bright image was in fact a blend of two images separated by∼ 0 5,

and thus very difficult to resolve with optical telescopes from the ground Thefact that the close pair is much brighter than the other two images is a genericprediction of lens theory, as will be shown below

Until 1990, a few more lens systems or lens candidate systems have beendiscovered, some of them from a systematic search for lenses amongst radiosources (e.g., Burke et al 1992), but most of them serendipitously (such asthe one shown in Fig 4) The 1990s then have witnessed several systematicsearches for lens systems, including programs carried out with the HubbleSpace Telescope (HST; Maoz et al 1993), lens searches amongst 15,000 radiosources (JVAS and CLASS; see King et al 1999; Browne et al 2003), andthose amongst very bright high-redshift quasars – these surveys will be de-tailed in SL (Part 2) By now, more than 80 multiple-image lens systems with

a galaxy acting as the (main) lens are known

Giant Luminous Arcs (1986)

In 1986, two groups (Lynds and Petrosian 1986; Soucail et al 1987) dently pointed out the existence of strongly elongated, curved features around

indepen-Fig 1 The two upper panels show a short (left ) and longer (right ) optical exposure

of the field of the double QSO 0957+561 (Young et al 1981) In the short exposure,the two QSO images are clearly visible as a pair of point sources, separated by∼6 .

The longer exposure reveals the presence of an extended source, the lens galaxy,between the two point sources, as well as a small cluster of galaxies of which thelens galaxy G1 is the brightest member The lower left panel shows a 6 cm VLAmap of the system (Harvanek et al 1997), where besides the two QSO sources Aand B, and the extended radio structure seen for image A, radio emission fromthe lens galaxy G is also visible The milli-arcsecond structure of the two compactcomponents A, B is shown in the lower right panel (Gorenstein et al 1988a), where

it is clearly seen that one VLBI jet is a linearly transformed version of the other,and they are mirror symmetric; this is predicted by any generic lens model whichassigns opposite parity to the two images

two clusters of galaxies (see Figs 5 and 6) Their tangential extent relative

to the cluster center was at least ten times their radial extent, although theexact value was difficult to determine as they were not well resolved in widthfrom the ground (HST has shown that this ratio is substantially larger than

Ly- b Damped system

Fig 2. Spectra of the twoimages of the lens systemQSO 0957+561, obtained withthe Faint Object Spectrograph

on board HST (Michalitsianos et

al 1997) The strong similarities

of the spectra, in particular thesame line ratios and the identicalredshift, verifies this system as adefinite gravitational lens system

10:1 in many cases) These giant luminous arcs were seen displaced from the

cluster center, and curving around it Various hypotheses were put forward

as to their nature, and all proven wrong, except for one (Paczy´nski 1987),when the redshift of the giant arc in A370 was measured (Soucail et al 1988)and shown to be much larger than the redshift of the cluster The arc wasthus proven to be a highly distorted and magnified image of an otherwise nor-mal, higher-redshift galaxy By now, many clusters with giant arcs are knownand have been investigated in detail As with most optical studies of lenses,the high-resolution of the HST was essential to study the detailed brightnessdistribution of arcs and to identify multiple images by their morphology and

colors Less distorted images of background galaxies have been named arclets

(Fort et al 1988); they can be identified in many clusters, and they are erally stretched tangentially with respect to the cluster center In addition,clusters can act as strong lenses also to produce multiple images of backgroundgalaxies Some of these aspects will be covered in Sect 4 of WL (Part 3)

gen-Rings, After All (1988)

Whereas Einstein ring images were predicted in the case of a perfectlycoaligned source with a spherically symmetric lens, the first multiple imageslens systems have taught us that lenses are far from spherical – thus, the dis-covery of a radio ring in the source MG 1131+0456 (Hewitt et al 1988) came

as a big surprise Unfortunately, owing to its faint optical counterpart, thelensing nature of this first system could not be proven easily, but the relative

Fig 3 In the left panel, a NIR image of the gravitational lens system PG 1115+080

is shown, taken with the NICMOS instrument on board HST The QSO has a redshift

of zs= 1.72 The double nature of the brightest component is clearly recognized, as well as the lens galaxy with redshift zd= 0.31, situated in the ‘middle’ of the four

QSO images When the QSO images and the lens galaxy are subtracted from the

picture, the remaining image of the system (right panel ) shows a nearly complete

ring, which is the lensed image of the host galaxy of the QSO, mapped onto a nearlycomplete Einstein ring In near-IR observations of lens systems, such rings occurfrequently (source: C Impey and NASA, see Impey et al 1998)

ease by which the radio source morphology, at several frequencies, could bemodeled by a simple gravitational lens (Kochanek et al 1989) made a verystrong case for its lensing nature The second radio ring discovered (Langston

et al 1989) made a much cleaner case: of the two radio lobes of a redshift 1.72quasar, one of them is imaged into a ring (see Fig 7) At the center of this

ring lies a bright, redshift zd= 0.25 galaxy, responsible for the light deflection.

High-resolution imaging with HST in optical and near-infrared filters revealedthe presence of Einstein rings in many multiply imaged quasars (Fig 8), wherethe host galaxy of the active nucleus is the corresponding (extended) source

We now know a lens needs not be exactly spherical; it is a combination ofthe asymmetry (ellipticity) of the mass distribution and the source size thatdetermines whether we will see an Einstein ring (see SL Part 2, Sect 10)

Fig 4 Around the center of this nearby spiral galaxy (zd= 0.04), four point-like

sources are seen is a fairly symmetric geometry (Yee 1988) Their spectra

iden-tify them as four images of a background QSO with zs = 1.7 This system, QSO

2237+0305, is the closest gravitational lens and one of the few systems where thelens is a spiral; it has been found in a spectroscopic redshift survey of nearby galaxies

Fig 5 The giant arc in the

cluster of galaxies Cl 2244−02,

taken with the ISAAC ment at the VLT (source: ESOPress Photo 46d/98) The arc

instru-has a redshift of zs = 2.24,

and was at the time of ery the highest redshift normalgalaxy The high magnificationcaused by the gravitational lensrenders this still (one of) the

discov-brightest galaxies with z ≥ 2

Quasar Microlensing (1989)

The mass of galaxies is not distributed smoothly, since at least a fraction of it

is in stars These stars will split the (macro)images of a quasar into many croimages whose typical separations of few micro-arcseconds are unresolvable.However, these perturbations of the gravitational field change the magnifica-tion of the macroimages, provided the source is sufficiently compact Since

mi-Fig 6 The cluster A2218 at z = 0.175 contains one of the most impressive systems

of arcs, as can be seen in the multi-color images taken with the WFPC2 instrument

on board HST (source: NASA/STScI) This cluster contains several multiple age systems of background galaxies which, together with the morphology of arcs,allows the construction of very detailed mass models for this cluster Also remark-able is the thinness of several of the arcs, so that they are not resolved in widtheven with the HST; this implies very large length-to-width ratios of these arcs and,correspondingly, very high magnifications

im-the source, im-the lens and im-the observer are not stationary, and im-the stars in im-thegalaxies move, this magnification will also change in time; the characteristictime-scales are of order a decade or less, and in one case (QSO 2237+0305, see

Fig 4) where the lens is very close to us (zd= 0.0395), even smaller Hence,

as predicted by Chang and Refsdal (1979, 1984), Paczy´nski (1986a), Kayser

et al (1986) and Schneider and Weiss (1987), this microlensing effect should

yield flux variations of the images which are uncorrelated between the differentimages – an intrinsic variation of the source would affect the flux of all images

in the same way, though with a time delay In 1989, this microlensing effectwas detected in the four image quasar lens QSO 2237+0305 as uncorrelatedbrightness variations in the four images (Irwin et al 1989)

Weak Lensing (1990)

As mentioned before, arclets are images of background galaxies stretched bythe lensing effect of a cluster In order to identify an arclet as such, the im-age distortion must be significant; otherwise, owing to the intrinsic ellipticitydistribution of galaxies, the stretching could not be distinguished from theintrinsic shape However, if the distortion field varies slowly with position,then galaxy images lying close to each other should be distorted by a similardegree Since we live in a Universe where the sky is densely covered with faintand small galaxies (e.g., Tyson 1988; Williams et al 1996), an average over

Fig 7 The quasar MG 1654+13 at redshift zs= 1.72 is shown, both as an optical

image (gray scale) and in the radio (contours) The optical QSO is denoted as Q,and is the central component (or core) of a triple radio source The Northern radiolobe is denoted by C, whereas the Southern radio lobe is mapped onto an Einsteinring At the center of this ring, one sees a bright galaxy with spectroscopic redshift

of zd= 0.25 This galaxy lenses the second radio lobe into a complete Einstein ring.

Within this ring, brightness peaks can be identified, and the components denoted Aand B are similar to, but not multiple images of, the brightness peak in the Northernlobe C (source: G Langston)

local ensembles of galaxies can be taken; the mean distortion of this ensemble

is then a measure for the lens stretching This weak gravitational lensing effect

was first detected in two clusters in 1990 (Tyson et al 1990) The advances

in optical imaging cameras, in particular the availability of large mosaic CCDcameras which enable the mapping of nearly degree-sized fields in a singlepointing, and the development of specific image analysis tools, have permit-ted the detection and quantitative analysis of weak lensing in many clusters.Even weaker lensing effects, those by an ensemble of galaxies and of the large-scale matter distribution in the Universe were discovered in 1996 (Brainerd

et al 1996) and 2000 (Bacon et al 2000; Kaiser et al 2000; van Waerbeke

et al 2000; Wittman et al 2000); we shall report on this in WL (Part 3)

Fig 8 The gravitational lens system B 1938+666 The left panel shows a

NIC-MOS@HST image of the system, clearly showing a complete Einstein ring intowhich the Active Galaxy is mapped, together with the lens galaxy situated near

the center of the ring The right panel shows the NICMOS image as gray-scales,

with the radio observations superposed as contours The radio source is indeed adouble, with one component being imaged twice (the two images just outside andjust inside the Einstein ring), whereas the other source component has four imagesalong the Einstein ring, with two of them close together (source: L.J King, see King

et al 1998)

Following Refsdal’s idea to determine the Hubble constant from lensing bycombining a good mass model for the lens with the time delay, the lightcurves of the first double QSO 095+561 were monitored by several groups

in the optical and radio waveband (e.g., Vanderriest et al 1989; Schild 1990;Leh´ar et al 1992) From these light curves, estimates of the time delay werederived by a number of groups, and significantly different results were ob-tained Difficulties include seasonal gaps in the optical light curves and thepossibility of uncorrelated variability in the images due to microlensing by thelensing galaxy To account for these effects, different methods were developed,yielding different results; broadly speaking, either delays of 410 days or 540days were obtained The issue was put to rest when a relatively sharp varia-tion of the flux of the leading image was detected in December 1994 (Kundi´c

et al 1995; Fig 9) Each of the two estimates for the time delay predicted adifferent epoch for the occurrence of the corresponding feature in the otherimage With the observation of the feature in the trailing image in February

1996 (Kundi´c et al 1997), the controversy was resolved in favor of the shortdelay, yielding 417± 3 days Time delays have now been measured in 10 lens

Fig 9 Light curves of the two

im-ages of the QSO 0957+561A,B in twodifferent filters The two light curveshave been shifted in time relative toeach other by the measured time de-lay of 417 days, and in flux accord-ing to the flux ratio The sharp dropmeasured in image A in Dec 1994and subsequently in image B in Feb

1996 provides an accurate ment of the time delay (data fromKundi´c et al 1997)

measure-systems, although the resulting estimates for the Hubble constant are stillproblematic – see SL (Part 2)

‘Jupiters’, black holes The ‘only’ problem was that about 1 out of 107 stars

in the LMC is expected to be lensed at any given time – the number ofstars needed to be monitored is indeed large Nevertheless, two groups startedthis adventure in the early 1990s, and reported in 1993 the first microlensingevents toward the LMC (Alcock et al 1993; Aubourg et al 1993) (Fig 10).Shortly thereafter, a third group announced the discovery of microlensingevents toward the Galactic bulge (Udalski et al 1993) Since then, this field hasflourished, and will be covered in depth in ML In addition to the discovery of

a large number of microlensing events, these surveys provide unique data setswhich are also useful for other branches of astronomy, most notably studies

of stellar statistics and variability

Fig 10 Blue and red light curve

of the first Galactic ing event MACHO-LMC-1 (Al-cock et al 1993) Data pointswith error bars show the mea-sured brightness of a star in theLMC as a function of time, andthe curve in both upper panelsshow the best fitting ‘standard’microlensing lightcurve Overall,the quality of the fit is impres-sive, and the lack of chromatic ef-fects, demonstrated by the con-stancy of the flux ratio shown

microlens-in the lowest panel, strongly gues for this being a microlens-ing event However, some points(in particular one close to themaximum flux) deviate very sig-nificantly from the simple modellightcurve, indicating that thismay be a binary microlens

ar-1.3 What is Lensing Good for?

Hopefully, by the end of these lectures we will have provided convincing swers to this question, but for the impatient, we shall summarize some of thehighlights of lensing applications

an-Measure Mass and Mass Distributions

Gravitational light deflection is determined by the gravitational field throughwhich light propagates This in turn is related to the mass distribution viathe Poisson equation (or its GR generalization) It is essential to realize that

this simple fact implies that gravitational light deflection is independent of

the nature of the matter and of its state – lensing is equally sensitive to dark

and luminous matter, and to matter in equilibrium or far out of it On thenegative side, this implies that lensing alone cannot distinguish between theseforms of matter, but on the positive side, it also cannot miss one of thesematter forms Hence, lensing is an ideal tool for measuring the total mass ofastronomical bodies, dark and luminous

From the Einstein deflection law (1), it is obvious that characteristic image

separations scale with the lens mass like M 1/2; hence, the observation ofmultiple images and rings immediately allows an estimate of the mass ofthe lensing galaxy – or more precisely, the mass within a cylinder with a

diameter of the image separation or the ring diameter, centered on the lens.2

More detailed modeling, and additional observables, such as flux ratios, canyield very precise mass estimates Indeed, as will be discussed in SL (Part2), accurate mass estimates within galaxies, with an uncertainty of a fewpercent, have been achieved – by far the most precise mass determinations

in (extragalactic) astronomy Similarly, from the locations of giant arcs inclusters, the masses of the central parts of clusters can be determined (Sect 4

of WL Part 3) With the advent of HST imaging and the discovery of multipleimage systems in some strong lensing clusters, detailed mass models have beenobtained, which led to very precise mass estimates in those clusters (needless

to say, they confirm the dominance of dark matter in clusters)

Weak lensing studies of clusters estimate the mass distribution to muchlarger radii than the strong lensing regime, and, like strong lensing effects,probe for asymmetries and substructures in the cluster mass For example,already the strong lensing properties of the cluster A2218 (Fig 6) reveals thebimodal nature of the mass distribution In fact, substructure in the massdistribution of lens galaxies has been detected, thereby confirming one of therobust predictions of the Cold Dark Matter model for our Universe (SL Part

2, Sect 8) In addition, the mass distribution of galaxies at large radii, whereone runs out of local dynamical tracers, can be studied statistically using aneffect called galaxy–galaxy lensing (WL Part 3, Sect 8)

Constraining the Number Density of Mass Concentrations

The probability for a lensing event to occur (e.g., the fraction of high-redshiftsources that are multiply imaged, or the fraction of stars undergoing mi-crolensing) depends on the projected number density of potential lenses.Hence, by investigating statistically well-defined samples of sources and theirlensed fraction, we can infer the number density of lenses Examples of suchstudies are estimates of the number density of compact objects in the darkhalo of our Galaxy, the redshift evolution of the number density of galaxiesacting as strong lenses, and the number density of clusters producing strongand weak lensing signals Upper limits on the number of lensing events canalso be translated into upper bounds on the number density of putative lenses:e.g., the fact that nearly all multiply-imaged sources have a visible lens galaxyputs strong upper bounds on the number density of dark lenses (they can atmost provide a few percent of the galaxy-mass objects), and the non-detection

of lens systems with image separations of tens of milli-arcseconds providesbounds on the number density of compact galaxies with masses ∼ 109M 

In fact, by now lensing has put stringent constraints on the population ofcompact massive objects in the Universe over an extremely broad range ofmass scales, from∼ 10 −3 M  (from upper limits on the variability of distant

2Whereas this ‘cylinder’ contains all the mass inhomogeneities of the cosmic matterdistribution between the source and the observer, it is dominated by the mass ofthe lensing galaxy

quasars) to∼1016M  (from the absence of very wide pairs of quasars), withonly a few mass gaps within this range Even lower-mass objects (∼10 −6 M )

can be ruled out as significant contributors to the dark matter in our MilkyWay (see ML)

Providing Estimates of Cosmological Parameters

Following Refsdal’s idea, the Hubble constant can be obtained from the timedelay in multiple image systems This method has the advantage of being inde-

pendent of the usual distance ladder used in determinations of H0, and it alsomeasures the Hubble constant on a truly cosmic scale, in contrast to the quitelocal measurements based on Cepheid distances Despite the determination

of time delays in a number of systems, values for H0 by lensing are burdenedwith the uncertainties of the lens models; however, there is a trend towardslightly lower values of the Hubble constant than obtained from Cepheids(see SL Part 2, Sect 5) Other cosmological parameters can also be obtainedfrom lensing For example, the fraction of lensed high-redshift quasars whencombined with the distribution of image separations can be used to estimatethe cosmological model (SL Part 2, Sect 6) Weak lensing by the large-scalestructure is sensitive to the matter density parameter and the normalization ofthe density fluctuations, and significant constraints on these parameters havebeen obtained (WL Part 3, Sect 7) In particular in combination with resultsfrom the anisotropy of the cosmic microwave background, future cosmic shearstudies will provide an invaluable probe of the equation of state of the darkenergy Weak lensing has also successfully been used to determine the biasparameter, which describes the relation between the statistical distribution ofgalaxies and the underlying dark matter, and for which only few alternativemethods are available (WL Part 3, Sect 8)

Lenses as Natural Telescopes

Since a lens can magnify background sources, these appear brighter than theywould without a lens This makes it easier to investigate these sources indetail, e.g through spectroscopic observations In some cases, this magnifica-tion is even essential to detect the sources in the first place, provided theirlensed brightness just exceeds the detection threshold of a survey or of thecurrent instrumental sensitivity This magnification effect has in fact yieldedspectacular results, such as very detailed spectra of very distant galaxies, thedetection of some of the highest redshift galaxies behind cluster lenses, andthe detection of very faint sub-millimeter sources in cluster fields.3 In fact,

3A magnification by a factor of, say, 5 implies that a spectrum of the source can

be taken in 1/25-th of the time it would take to get the same signal-to-noisespectrum of the unlensed source Needless to say that such a factor can make thedifference between an observation being made and one that cannot be done

Fig 11 Example for the use of a gravitational lens as a natural telescope In a

search for very high redshift objects, deep multi-band HST images are taken nearthe critical curves of clusters, where large magnifications are expected Shown hereare images of a field in the cluster A2218 (see Fig 6) in four filters, ranging from

0.6 μm to the near-IR at 1.6 μm In the two larger wavelength images, a double source

is seen, which is absent at shorter wavelength The two components are situated at

opposite sides of the critical curve, which is drawn for three source redshifts of zs= 6,6.5 and 7; due to the large number of strong lensing constraints for this cluster, itsmass distribution in the central part is very well determined The sticks indicatethe shear field of the cluster, and the elongation of the double images is parallel tothis shear, as expected if they were gravitationally lensed images From the location

of the images with respect to the critical curve, and the drop-out of their flux atwavelengths shorter than ∼ 0.8 μm, the redshift of the source is estimated to be between zs= 6.5 to 7 (from Kneib et al 2004)

the lens magnification can be very large in some rare cases, but these rarecases truly stick out: some of the most extreme sources, with regards to theirapparent luminosity, are strongly magnified – such as the spectacular IRASgalaxy F10214 (e.g., Broadhurst and Leh´ar 1995), the by-far brightest redshift

∼3 galaxy cB58 (Seitz et al 1998), or the extremely luminous z = 3.87 quasar

QSO APM 08279+5255 (Irwin et al 1998).4 A good fraction of known ies with redshift larger than ∼ 4 have been detected behind cluster lenses,

galax-including the redshift record holder candidate (z = 10.0) at present (Pell´o

et al 2004); see Fig 11 for an example During high-magnification tic microlensing events, detailed spectra of stars at large distances (e.g theGalactic bulge) have been taken As the high magnification region crosses adistant star, observations have mapped the surface brightness distribution ofthe stars to test stellar atmosphere models

Galac-With the lenses as magnifiers, larger effective angular resolution of thesources is obtained Galaxies acting as sources for giant arcs can therefore beresolved in unprecedented detail, at least in one dimension The host galaxy

of quasars, which is difficult to study in unlensed objects owing to the largebrightness contrast between the active nucleus and the surrounding host, can4

Such extremely bright quasars are of great importance for the study of the

inter-galactic absorption, e.g., the Ly-α forest; no surprise then that such objects, like the highly magnified z = 3.62 QSO 1422+231, are preferred targets for investi-

gating absorption lines

be studied much more easily when lensing allows the spatial resolution of thehost – in many cases, the host galaxy is in fact mapped into an Einstein ring(see Figs 3 and 8).

Searches for Planets

The light curves of Galactic microlensing events are affected by companions ofthe main lens For example, light curves of binary stars are readily identified

as such, provided their separation falls into a favorable range determined bythe geometry of the lens system Because of that, even planets will leave anobservable trace in the microlensing light curves if they are situated at theright radius from the star and at the right orbital phase Although these tracescan be quite subtle, and last for a short time only, current observing campaignsaimed at the search for planets have the sensitivity for their detection, andseveral candidate events for the detection of planetary signals in microlensinglight curves have been reported Indeed, microlensing is considered to be thesimplest (and cheapest) possibility to detect the presence of low-mass planetsaround distant stars (ML)

These few examples should suffice to illustrate the broad range of cations of gravitational lensing; the ever increased publication rate of articlesinvestigating and applying gravitational lensing underlines the timeliness ofthe subject

appli-2 Gravitational Lens Theory

Assuming the validity of General Relativity, light propagates along the nullgeodesics of the space–time metric However, most astrophysically relevantsituations permit a much simpler approximate description of light rays, which

is called gravitational lens theory In this section, we summarize the basicequations for the description of light deflection in a gravitational field Thereader is referred to SEF and PLW for a more detailed account and furtherreferences

2.1 The Deflection Angle

Consider first the deflection of a light ray by the exterior of a spherically

symmetric mass M Provided that the ray impact parameter ξ is much larger than the Schwarzschild radius of the mass, ξ  Rs≡ 2GM c −2, then General

Relativity predicts that the deflection angle ˆα is

ˆ

α = 4GM

This is just twice the value obtained in Newtonian gravity (see Sect 1.1)

According to the condition ξ  R , the deflection angle is small, ˆα  1 This

condition also implies that the Newtonian gravitational field strength is small,

φN/c2 1.

The field equations of General Relativity can be linearized if the tional field is weak The deflection angle of an ensemble of mass points is thenthe (vectorial) sum of the deflections due to the individual mass components

gravita-A three-dimensional mass distribution with volume density ρ(r) can be vided into cells of size dV and mass dm = ρ(r) dV Let a light ray pass this

di-mass distribution, and describe its spatial trajectory by (ξ1(λ), ξ2(λ), r3(λ)),

where the coordinates are chosen such that the incoming light ray (i.e., far

from the deflecting mass distribution) propagates along r3 The actual lightray is deflected, but if the deflection angle is small, the ray can be approxi-mated as a straight line in the neighborhood of the deflecting mass (note thatthis corresponds to the Born approximation in atomic and nuclear physics) A

mass distribution for which this condition is satisfied is called a

geometrically-thin lens Then, ξ(λ) ≈ ξ, independent of the affine parameter λ Note that

ξ = (ξ1, ξ2) is a two-dimensional vector The impact vector of the light ray

relative to the mass element dm at r  = (ξ1 , ξ 2, r 3) is then ξ −ξ , independent

of r 3, and the total deflection angle is

dent of r3 , the r3 -integration can be carried out by defining the surface mass

density

Σ(ξ) ≡



dr3ρ(ξ1, ξ2, r3) , (4)which is the mass density projected onto a plane perpendicular to the in-coming light ray Thus, the deflection angle produced by an arbitrary densitydistribution is

sky This angle, however, is typically very small (in the case of cluster lensing,

the relevant angular scales are of order 1 arc min≈ 3 × 10 −4radians).

ObserverLens plane

Source plane

qbx

Fig 12 Sketch of a typical gravitational lens system

2.2 The Lens Equation

A typical situation considered in gravitational lensing is sketched in Fig 12,

where a mass concentration at redshift zd (or distance Dd) deflects the light

rays from a source at redshift zs(or distance Ds) If there are no other tors close to the line-of-sight, and if the extent of the deflecting mass along the

deflec-line-of-sight is very much smaller than both Ddand the distance Ddsfrom thedeflector to the source,5 the actual light rays which are smoothly curved inthe neighborhood of the deflector can be replaced by two straight rays with akink near the deflector The magnitude and direction of this kink is described

by the deflection angle ˆα, which depends on the mass distribution of the

de-flector and the impact vector of the light ray The lens equation relates thetrue position of the source to its observed position on the sky As sketched inFig 12, the source and lens planes are defined as planes perpendicular to astraight line (the optical axis) from the observer to the lens at the distance

5This condition is very well satisfied in most astrophysical situations A cluster of

galaxies, for instance, has a typical size of a few Mpc, whereas the distances Dd,

D , and D are fair fractions of the Hubble length cH −1 = 3 h −1 × 103Mpc

of the source and of the lens, respectively The exact definition of the cal axis does not matter because of the smallness of the angles involved in a

opti-typical lens system Let η denote the two-dimensional position of the source

on the source plane, measured with respect to the intersection point of theoptical axis with the source plane From Fig 12 we can read off the geometriccondition that (again making use of the smallness of angles occurring, so thatsin ˆα ≈ ˆα ≈ tan ˆα)

inter-be seen by an observer to inter-be located at angular positions θ satisfying (8).

If (8) has more than one solution for fixed β, a source at β has images at

several positions on the sky, i.e., the lens produces multiple images For this

to happen, the lens must be ‘strong’ We can express the scaled deflectionangle in terms of the surface mass density as

which has κ ≥ 1 somewhere, i.e., Σ ≥ Σcr, produces multiple images for some

source positions Hence, Σcr is a characteristic value for the surface massdensity which is the dividing line between ‘weak’ and ‘strong’ lenses.6

The lens equation (8) describes a mapping θ → β from the lens plane

to the source plane; for any mass distribution Σ(ξ), this mapping can (in

principle) be easily calculated One problem of gravitational lens theory is

the inversion of (8), i.e., to find all the image positions θ for a given source

6In order to derive the foregoing equations, we have used Euclidean geometry

to relate angles to length scales We shall discuss in Sect 4 that the equations

still hold in an expanding universe, provided the distances D’s are interpreted as angular diameter distances – hence, in the notation of Sect 4, D ≡ Dang

position β Since the mapping θ → β is non-linear, the inversion of the lens

equation can be carried out analytically only for very simple mass models of

the lens As the number of images θ for a given source β is not known a priori,

a numerical inversion is non-trivial in general; however, we shall see below thatthere are methods to determine the image multiplicity as a function of thesource position

The identity ∇ ln |θ| = θ/|θ|2, valid for any two-dimensional vector θ,

shows that the (scaled) deflection angle can be written as a gradient of the

so that the mapping θ → β is a gradient mapping Furthermore, using the

identity ∇2ln|θ| = 2πδD(θ), where δD is the (two-dimensional) Dirac delta

‘function’, one obtains from (11) that

which is the Poisson equation in two dimensions The similarity between these lensing relations and standard three-dimensional gravity (ψ corresponds to the gravitational potential φN, α corresponds to the acceleration vector, κ

corresponds to the volume mass density ρ) shall be noted.

For later purposes, we shall find it useful to define a further scalar function

is equivalent to the lens equation (8) As has been shown in Schneider (1985);

see also SEF), the function τ (θ; β) is, up to an affine transformation, the light travel time along a ray starting at position β, traversing the lens plane

at position θ and arriving at the observer Thus, (15) expresses the fact that

physical light rays are those for which the light travel time is stationary –which thus expresses Fermat principle in the context of lensing by a geomet-rically thin matter distribution We shall see that the Fermat potential – ortime-delay function – is very useful for a classification of the multiple images

in a gravitational lens system Displaying lens properties in terms of the mat potential (Blandford and Narayan 1986) provides useful insight in thebehavior of the lens mapping

Fer-2.3 Magnification and Distortion

The solutions θ of the lens equation yield the angular positions of the images

of a source at β The shapes of the images will differ from the shape of the

source because light bundles are deflected differentially, as we saw from theimages of giant arcs in Fig 6 In general, the shape of the images must be de-termined by solving the lens equation for all points within an extended source.Liouville theorem and the absence of emission and absorption of photons ingravitational light deflection imply that lensing conserves surface brightness

(or specific intensity) Hence, if I(s)(β) is the surface brightness distribution

in the source plane, the observed surface brightness distribution in the lensplane is

and κ is related to ψ through Poisson equation (13) Hence, if θ0 is a point

within an image, corresponding to the point β0 = β(θ0) within the source,

we find from (16), using the locally linearized lens equation,

I(θ) = I(s)[β0+A(θ0)· (θ − θ0)] (19)According to this equation, the images of a source with circular isophotesare ellipses The ratios of the semi-axes of such an ellipse to the radius ofthe source are given by the inverse of the eigenvalues of A(θ0), which are

1− κ ± |γ|, and the ratio of the solid angles subtended by an image and the

unlensed source is the inverse of the (absolute value of the) determinant ofA.

The inverse of the Jacobian is called the magnification tensor,

and yields the local mapping from the source to the image plane The fluxesobserved from the image and from the unlensed source are given as integrals

over the brightness distributions I(θ) and I(s)(β), respectively, and their ratio

is the magnification |μ(θ0)| From (19), we find for the magnification of a

‘small’ source

μ = det M = 1

detA =

1(1− κ)2− |γ|2 . (21)

The images are thus distorted in shape and size The shape distortion is due to

the tidal gravitational field, described by the shear γ, whereas the tion is caused by both isotropic focusing due to the local matter density κ and

magnifica-anisotropic focusing due to shear The magnification as defined in (21) can

have either sign; the sign of μ is called the parity of an image Negative-parity

images are mirror-symmetric images of the source Of course, the observed

fluxes of images are determined by the absolute value of μ Since the intrinsic

luminosity of sources is unknown, the magnification in a lens system is not anobservable However, the flux ratio of different images provides a direct mea-surement of the (absolute value of the) corresponding magnification ratio Ingeneral, if two extended images of a source are observed, then their shapesdepend on the shape of the source throughA As the shape of the source is

unknown, what can be determined from the shape of extended images is therelative magnification matrix A ij = A(θ i)A −1 (θ

j), which provides the earized mapping of one image onto the other Note thatA ij is in general notsymmetric and thus has four independent components For a pair of imageswith opposite parity, detA ij < 0, and so these two images are mirror sym-

lin-metric; an example of this can be seen in the VLBI images of QSO 0957+561(see Fig 1)

To consider the distortion of the shape of images in somewhat more detail,

we shall rewrite the Jacobian in a slightly different form,

As can be easily seen from (3), the factor (1− κ) only yields an isotropic

stretching of the image, but does not affect its shape The reduced shear

g – like γ – is considered to be a complex number, g = g1 + ig2 and itscomponents determine the change of shape between the source and the image

In particular, a circular source of unit radius is mapped onto an ellipse withaxes |(1 − κ)(1 + |g|)| −1 and |(1 − κ)(1 − |g|)| −1, and the orientation of the

ellipse is determined by the phase ϕ of g As will be seen in WL (Part 3), the

reduced shear is the central quantity in weak gravitational lensing

The images of a small source (what that means depends on the context;see below) are therefore magnified by|μ(θ i)|, and the total magnification of

a small source at position β is given by the sum of the magnifications over all

where I(s)(β) is the surface brightness profile of the source Whereas

grav-itational lensing is achromatic, because the deflection of photons does notdepend on their frequency, the finite resolution of observations can lead to

color terms in practice, since the surface brightness distribution I(s)(β) can

be different at different frequencies Then, if the magnification μp(β) varies

on scales comparable to the source size, the magnification of an extended but

unresolved source can depend on the frequency.

Since the shear is defined by the trace-free part of the symmetric bian matrixA, it has two independent components There exists a one-to-one

Jaco-mapping from symmetric, trace-free 2× 2 matrices onto complex numbers,

and we shall extensively use complex notation Note that the shear (and thereduced shear) transforms as e2iϕ under rotations of the coordinate frame,

and is therefore not a vector (but a polar, i.e., it has the same transformation

properties as the linear polarization of electromagnetic waves) Equations (11)and (18) imply that the complex shear can be written as

2.4 Critical Curves and Caustics, and General Properties of Lenses

In any lens there can be closed, smooth curves, known as critical curves, on

which the Jacobian vanishes, detA(θ) = 0 The curves in the source plane

which are obtained by mapping the critical curves with the lens equation are

called caustics, which are not necessarily smooth, but can develop cusps

Criti-cal curves and caustics are of great importance for a qualitative understanding

of the lens mapping, owing to their following properties:

1 The magnification μ = 1/ det A formally diverges for an image on a critical

curve Infinite magnifications are of course unphysical All astronomicalsources have a finite size that keeps their observed magnification (25)finite For a hypothetical source of vanishing extent, the magnificationwould be finite because the geometrical optics approximation then breaksdown and we must use wave optics The resulting diffraction patterns pre-dict finite, though potentially very high magnifications (see e.g Ohanian

1983 or Chap 7 of SEF) Nevertheless, a source located near a caustic canproduce very highly magnified images close to the corresponding criticalcurve in the lens plane

2 The number of images a source produces depends on its location relative

to the caustic curves Assuming a mass profile of a lens for which the

deflection angle tends to zero for large|θ| – as is true for all real lenses

– and an upper bound to the deflection angle (i.e., excluding point-masslenses for the moment), a source at large|β| will have only one image, at

θ ≈ β, whereas it can have multiple images for small impact vectors The

lens mapping (8) is locally invertible at all locations for which detThis immediately implies that a change of the source position does not

lead to the change of the number of images unless the source moves across

a caustic – since caustics are obtained by mapping the critical curves(where the lens mapping in not invertible) onto the source plane When asource position crosses a caustic, a pair of images near the correspondingcritical curve is either created or destroyed, depending on the direction

of crossing The side of the caustic where the number of images is larger

by two is often called the ‘inner side’ A source close to, and on the innerside of a caustic possesses a pair of images with very high and nearlyequal magnification on either side of the critical curve, in addition to anyother images The bright pair must have opposite parities because themagnification changes sign at the critical curve

Whereas the critical curves are smooth, this does not need to be the case

for caustics To see that, let θ(λ) be a parameterization of a critical curve; the caustic then is β(θ(λ)) The tangent vector to the critical curve is the derivative ˙θ(λ) ≡ dθ(λ)/dλ, and the tangent vector to the caustic is

This vector, however, can vanish if the tangent vector to the critical curve ˙θ(λ)

is parallel to the eigenvector of A whose eigenvalue is 0 (remember that we

are analyzing a critical curve, along which one eigenvalue ofA is always zero).

Hence, if the direction of the tangent vector to the critical curve is the singulardirection of A, the caustic curve need not be smooth; in fact, it has a cusp.

Apart from any cusps the caustic curves are smooth curves called fold caustics.

These names are taken from singularity theory, a mathematical discipline thatstudies the critical points of general mappings We shall see the occurrence

of cusps later in several specific examples of lens mappings A source close toand inside a cusp has three highly magnified images near the correspondingpoint on the critical curve; one can show (see e.g Schneider and Weiss 1992;Mao 1992) that the sum of the absolute values of the magnification of the twoouter images equals the absolute value of the magnification of the central ofthese three images A source just outside the cusp has one highly magnifiedimage near the corresponding critical curve

We thus obtain a qualitative understanding of the geometry of a lensmapping from the critical curves and caustics The critical curves divide the

lens plane into regions of positive (i.e., μ > 0) and negative (μ < 0) parity The

corresponding caustics divide the source plane into regions of different imagemultiplicity: whenever a source position changes across a caustic, the number